python controller module code works inside blender but not saved as game engine runitime
Monster's hint pointed me towards the real answer: Always look at the console window, when you are working with python. 
(HowTo is described here: https://wiki.blender.org/index.php/User:Fade/Doc:2.6/Manual/Interface/Window_system/Console_window .) When I was using the Embedded Player in the Blender Game Properties Editor, the console printed out that it not can "import bpy". 
So I realized that I only have to "import bge" and not have to "import bpy".

The second part to the answer I found in MikePan's answer in this post: 
Packaging external libraries in the game engine &#39;Create Runtime&#39;. I work on a Mac and I simply had to copy the python file withe code for the controller into the Content/MacOS/Resources folder of the Save As Game Engine Runtime generated app. 

This question and comments are as well related to the topic: BGE: where to place external python modules when exporting for Mac OS X